MATH 222 / 472

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Item No. 1

[13 Marks]

Consider the following DE for items a) through f) below:

dt= ln t t ln V, t > 0, V > 0

a) Is the DE separable? Provide justification for your answer.

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b) Is the DE linear? Provide justification for your answer.

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c) Is the DE of Bernoulli type? Provide justification for your answer.

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d) Is the DE exact? Provide justification for your answer.

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e) Is the DE homogeneous in the sense that dV

dt= F (V /t)? Provide justification for your answer.

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f) Suggest an appropriate method to solve the DE, and then solve it. Present all steps clearly.

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Item No. 2

[8 Marks]

Consider the DE

2t cos T

dt=

t sin T + 2T

Examine this DE and determine whether it is separable, linear, or exact, providing justification

for each classification. Based on your investigation, select an appropriate method and solve the DE,

showing all steps clearly.

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Item No. 3

[7 Marks]

Let F denote the set of all real-valued functions:

F= (f : R ! R

f is a real-valued function)

Given f and g in F and a real number r 2 R, the functions f + g and rf are defined in the natural way,

(f + g)(x) = f(x) + g(x)

and

(rf)(x) = r(f(x))

Verify that F is a vector space.

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Item No. 4

Let P2 denote the set of all polynomials in x of degree at most 2:

P2 =

8 < :

f (x) = ax2 + bx + c,

a, b, c 2 R,

deg(f ) ? 2

9 = ;

For f1, f2 2 P2, define addition by the sum of corresponding terms, and for any scalar k 2 R, define

scalar multiplication by multiplying each term of f by k.

Use the information above to answer items a) through c) below.

a) Show that the set P2, equipped with the above operations, is a subspace of the vector space F.

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Let f1(x) = 1727, f2(x) = 1932x, and f3(x) = 1938×2

. Show that the set {f1, f2, f3} is linearly

independent and that it spans the entire vector space P2. Finally, write

f (x) = 1980×2 + 2016x + 2030

as a linear combination of f1, f2, and f3.

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c) What is the dimension of P2?

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Item No. 5

[6 Marks]

Consider the linear first-order homogeneous DE dy

dx + p(x)y = 0.

Examine the properties of its solutions and answer the following:

a) Verify whether the function y = 0 satisfies the given differential equation. Explain your reasoning

in terms of the definition of a solution.

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b) Suppose y1 = f1(x) and y2 = f2(x) are two solutions of the DE, and r 2 R. Show that both

y1 + y2 and ry1 are solutions. Justify your conclusion using the linearity property of the differential

operator.

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c) Define the set of solutions of dy

dx + p(x)y = 0 by S= nf (x)

your answers in parts a) and b), conclude that S is a subspace of F.

dx [f (x)] + p(x)f (x) = 0o. Based on

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Item No. 6

[9 Marks]

Consider the following vectors in R3 to answer items a) through b) below:

v1 =

2 6 6 4

3 7 7 5, v2 =

2 6 6 40

3 7 7 5, v3 =

2 6 6 4

3 7 7 5, v4 =

2 6 6 40

3 7 7 5

a) Determine whether the set {v1, v2, v3, v4} is linearly independent or dependent. Explain your

reasoning based on the definition of linear dependence.

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b) Express the vector v1 as a linear combination of v2, v3, and v4.

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Item No. 7

[12 Marks]

Consider the homogeneous linear system to answer items a) through f) below:

x1 4×2 3×3 7×4 = 0

2×1 x2 + x3 + 7×4 = 0

x1 + 2×2 + 3×3 + 11×4 = 0.

a) Find an echelon form of the coefficient matrix.

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b) The system has 4 variables and 3 equations. What is the maximum number of leading

variables the system can have?

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c) What does this imply about the number of free variables?

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d) Find a basis for the solution space.

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e) Find the dimension of the solution space.

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f) State the relationship between the number of free variables and the dimension of the solution

space.

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